Elliptic deformations of current algebras and their representations by difference operators (Q1128369)

The ``current'' realization of an affine quantum group was constructed by V. G. Drinfeld. This realization contains a subalgebra of deformations of the universal enveloping algebra of the Lie algebra of currents on a circle. Let \(\Delta\) be the root system of a finite-dimensional semisimple Lie algebra \(\mathfrak g\) and let \(n: L\to\mathbb{Z}\) be a strictly dominant integral form on the integral lattice \(L\) generated by the simple roots of algebra \(\mathfrak g\). The paper deals with the associative algebras \(Q_{n,\Delta}(E,\tau)\). Here \(\tau\in\mathbb{C}\) and \(E=\mathbb{C}/\Gamma\) is an elliptic curve, \(\Gamma=\{m_1+m_2\eta; m_1,m_2\in\mathbb{Z}\}\), \(\Im \eta>0\). The authors construct a family of representations of the algebras \(Q_{n,\Delta}(E,\tau)\) and study the center of these algebras for the generic \(\tau\). The algebra \(Q_{n,\Delta}(E,\tau)\) can be interpreted as algebra of screening operators and is the subalgebra of the algebra \(F\) of ``screening'' currents. The algebra \(F\) can be realized by vertex operators inside the universal enveloping algebra of a Heisenberg algebra.